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Thursday, March 24, 2016

Calling All Number Theorists....

Those persnickety primes... Earlier today Mike Lawler posted some results from Mathematica for the patterns in last digits of prime-triples, essentially in intervals of a billion primes (not sure if I'm stating that very clearly, but read his post):


His columns are a bit hard to read, but you should be able to spot the "clustering" (or I would call it "coupling") he refers to which seems to largely hold for all 3 columns of the post. He notes in a comment that the "average" expected value for each entry would be about 15.6 million, so you can see how widely the values diverge from that, as well as see how they tend to pair up.
He is in the process of adding more columns at the below easier-to-read spreadsheet -- I assume he'll be going out to 10 billion primes, to restore data he originally had, but lost -- as I write this, columns for 4 billion primes are listed, and it appears to me (merely eyeballing it), that the numbers for the paired triplets are getting even closer(???): 


 As each set of a billion primes is a somewhat independent and random-like group of integers, this pattern of the same ordered-triplet of last digits re-occurring in associated pairs seems, on the surface at least, rather odd and striking!?  What (if anything) is it about those pairs? Perhaps a number theorist can see through to a simple explanation for it (if so, I'm sure Mike would love to hear it). Or does this finding piggy-back in any way on to the peculiar result from a week prior of prime number last digits tending to avoid repetition in consecutive primes?
WHAT is going on here....?

ADDENDUM:  I should have included in this post that Mike has already recognized that the paired triplets involved are consistently of the form (a, b, c) and (-c, -b, -a) in mod 10. Now THAT surely must mean something! (Again, perhaps something obvious to a number theorist, but WHAT?)

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